The Length of the Graph of a One to One Function from [0,1] to [0,1]
The Length of the Graph of a One to One Function from [0,1] to [0,1]
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从 [0,1] 到 [0,1] 的一对一函数图的长度
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发表时间：
1999
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通讯作者：
J. Foran
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作者：
J. Foran
上限和下限的长度的图形的一个到一个，上，Baire一个功能，从单位间隔本身被证明是无穷大和一个，分别。这类函数具有图的大尺寸和不可测的功能也被认为是。对于平面的子集E的长度，我们将使用一维Hausdorff测度Λ（E）= lim δ → 0 inf ∑ diam（Ei），其中下确界是在所有{Ei} i = 1上，其中E ≠ Ei且diam（Ei）<δ。我们需要的一些众所周知的事实是：1。若E是[a，B]上的连续函数的图，则Λ（E）与E的长度一致. 2.如果proj θ（E）是E在与x轴成θ角的直线上的垂直投影，则Λ（proj θ（E））≤ Λ（E）。我们最终也将使用E的s维Hausdorff测度，由Λ s（E）= lim δ → 0 inf ∑（diam（E））给出，其中下确界如上所述，当E是可测的时，Λ s（E）代替Λ s（E）。E的维数将意味着Hausdorff维数，dim（E）= inf {s：Λ s（E）= 0}。s维测度可以通过使用一系列规则网中的正方形来估计，我们最终将使用单位正方形上的网，其正方形的顶点坐标为m/K。参见例如，[1]或[4]。对于取[0，1]到[0，1]上的连续的一对一函数f，众所周知，f必须是单调的，这样的f的图的长度可以是[02，2]中的任何值。长度必须至少为1002是由于以下事实
Upper and lower limits for the length of the graph of one to one, onto, Baire one functions from the unit interval to itself are shown to be infinity and one, respectively. Such functions having graphs of large dimension and non-measurable functions are also considered. For the length of a subset E of the plane, we will use the one dimensional Hausdorff measure Λ(E) = lim δ→0 inf ∑ diam(Ei) where the infimum is over all {Ei}i=1 with E ⊂ ∪Ei and diam(Ei) < δ. Some well known facts which we will need are: 1. If E is the graph of a continuous function on [a, b] then Λ(E) agrees with the length of E. 2. If projθ(E) is the perpendicular projection of E onto a line making an angle θ with the x-axis then Λ(projθ(E)) ≤ Λ(E). We will eventually also use the s-dimensional Hausdorff measure of E given by Λs(E) = lim δ→0 inf ∑ (diam(E)) where the infimum is as above and Λs(E) replaces Λs(E) when E is measurable. By the dimension of E will be meant the Hausdorff dimension, dim(E) = inf{s : Λs(E) = 0}. The s-dimensional measure can be estimated by using squares from a sequence of regular nets and we will eventually use nets on the unit square whose squares have vertices with coordinates m/K. See e.g., [1] or [4]. For continuous one to one functions f taking [0, 1] onto [0, 1] it is well known that f must be monotone and the length of the graph of such an f can be any value in [ √ 2, 2]. That the length must be at least √ 2 is due to the fact